Why didn't Lord Voldemort want to kill Lily Potter?

Answer 1

Severus Snape, who always loved Lily, asked Voldemort to spare her. As Snape was (at the time) loyal to him, Voldemort decided to give her a chance but when she was unwilling to step away he killed her without another though.

Answer 2

Lord Voldemort tried to kill Harry Potter because Snape heard Trelawney's prophecy that said that a boy born on the last day of July whose parents had escaped him three times would have the power to kill him (Dumbledore also heard this prophecy) but Snape didn't hear all of it. When he told Voldemort, there were two boys who fit this description - Harry Potter and Neville Longbottom.

Either of these could have been the boy and the only reason it ended up being Harry was because Voldemort was unaware that the second part of the prophecy stated Voldemort would mark him as his equal. It is believed that Voldemort chose Harry as they were both half-bloods and he felt most threatened by him.

Voldemort could have avoided the problems had he not tried to kill Harry. When he tried to kill Harry and couldn't (because Harry's mother died to save his life), he actually gave Harry the ability to kill Voldemort.

He was also marked as Voldemort's equal more than before because some of Voldemort's powers were transferred and he became the owner of the wand with a twin core.

Voldemort never actually discovered what the prophecy said because it was destroyed when his death eaters broke into steal it. So he kept trying to finish what he had started believing that neither could live while the other survived, as the prophecy predicted.

Answer 3

He didn't really care if he killed her or not but when he got there he remembered that Severus Snape has begged him to spare her. Since Snape had given him the information that (he thought) would save him he was willing to do him a favour but when she refused to move he didn't think twice about murdering her.